Homogeneous locally nilpotent derivations of k[X, Y, Z] Original Research Article

Let k be a field of characteristic zero and let B = k[X,Y,Z] be a polynomial ring in three variables. A derivation D:B → B is said to be locally nilpotent if for each bεB we have Dn(b) = 0 for n much greater-than 0. This paper describes the class of homogeneous locally nilpotent derivations of B, where homogeneity is relative to any N-grading of B which is obtained by assigning weights to the variables, w(X) = a, w(Y) = b and w(Z) = c, with a, b, c ε N and gcd(a, b, c) = 1. It is known that the kernel of a derivation from this class is a subalgebra k[f,g] of B = k[X, Y, Z], where f and g are w-homogeneous and algebraically independent, and it is also known that such a derivation is essentially determined by its kernel. The main results of this paper give explicit conditions on f, g ε k[X, Y, Z] which are equivalent to k[f,g] being the kernel of a homogeneous locally nilpotent derivation of k[X, Y, Z].